Fluctuations of work in realistic equilibrium states of quantum systems with conserved quantities

We very much appreciate the Referee’s positive outlook on our paper, and have modified our manuscript in line with his comments. The changes are listed below. Let us here just briefly comment on two points:

• With regards to point 2 by the Referee, he is right in pointing that the eigenstates should generally be associated with a series of quantum numbers, one for each charge (Hamiltonian and other conserved charges). With this in mind, we have replaced |n> with | n, i_1, ... i_{N_cons} >, and indicate explicitly “n stands for the quantum number that identifies the energy eigenvalue, E_n, while i_k is the quantum number labelling the eigenvalues, M_{k, i_k }, of the charge operator \hat{M}_k.”, and similarly for the final states |m’> (lines 104-108).

• On the Referee's comment #3: Our generalised quantum fluctuation relations (as is generally the case with these kind of equalities) are valid for any time dependence in H(t). This generality includes processes where H(t) is parameterised by a continuous parameter, say H = H(lambda(t)) with lambda(t) a continuous function of t. But the QFRs can be also applied to quantum quenches, i.e., sudden changes of the parameters of the Hamiltonian. This is the case we have analysed in our simulations where the parameters (alpha, g) are changed in a step-wise manner. We now provide explicit details on this point when describing our numerical protocol in Section 3.2, and have added a more general statement in Sec. 2 (step 2 of the generalised TPM protocol).

We hope that, with these changes, our manuscript will be accepted for publication in SciPost Physics Proceedings.

1. We have added Refs. [1-4] to address the Referee’s comment #1.

2.In reply to the Referee’s comment #2: We have clarified the notation of the initial and final eigenstates, to account for the various conserved charges at each stage of the protocol.

3. In reply to the Referee’s comment #3: We provide a more detailed description of the time dependence of our Hamiltonian, with a new sentence in Section 2 (step 2 of the generalised TPM protocol), and a new short paragraph including a new equation specifying the functions {alpha(t), g(t)} we used in Section 3 (step 2 of the numerical protocol).

4. We corrected the text as requested by the Referee’s comment #4. The set of references [22, 34-36, 38] at the end of that sentence contains the further details on the existence of the additional conserved charge for the various values of alpha. We have also removed the subindex of M_k in line 152: as there is a single charge, there is no need to add a counter to distinguish it from others as in the general case.